# Tight Bounds for Asymptotic and Approximate Consensus

## TL;DR

This paper establishes tight lower bounds on the convergence rates of asymptotic and approximate consensus algorithms in dynamic networks, demonstrating their optimality and exploring their relation to exact consensus.

## Contribution

It provides the first tight bounds on contraction rates for consensus algorithms in dynamic networks, including the weakest solvable models, and introduces an extended valency concept.

## Key findings

- Proved tight lower bounds on contraction rates.
- Demonstrated optimality of existing algorithms.
- Linked topological properties of valencies to consensus solvability.

## Abstract

We study the performance of asymptotic and approximate consensus algorithms under harsh environmental conditions. The asymptotic consensus problem requires a set of agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values. This problem, and the related approximate consensus problem, are fundamental building blocks in distributed systems where exact consensus among agents is not required or possible, e.g., man-made distributed control systems, and have applications in the analysis of natural distributed systems, such as flocking and opinion dynamics. We prove tight lower bounds on the contraction rates of asymptotic consensus algorithms in dynamic networks, from which we deduce bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented in [Charron-Bost et al., ICALP'16] for certain dynamic networks, including the weakest dynamic network model in which asymptotic and approximate consensus are solvable. As a corollary we also obtain asymptotically tight bounds for asymptotic consensus in the classical asynchronous model with crashes. Central to our lower bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks.